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192 Niels J. Olieman and Eligius M.T. HendrixAny of the enclosed or enclosing shapes can be used for the computation of respectively arobustness lower bound or upper bound. Since c is fixed, the radius of any of the shapes, isproportional to the corresponding probability bound. Thus the comparison of the bounds,can be based on the radius in combination with property (8).The challenges for GO are two-fold:and1. For the maximisation and minimisation problems inside the definitions (9) and (10), it iscrucial to find global optima, since only in that situation a bound can be guaranteed.2. The R(x)-bounding approach is competing with R(x)-estimation. In the situation aMonte Carlo sampling approach for estimating R(x) and N samples are sufficient forestimating R(x) at a satisfying accuracy level, then it only makes sense if the boundingapproach can find a bound within < N function evaluations. In practice this means fastglobal optimisation routines are needed.As an illustration see figure 1 whereu(x, v) = 2 − (x 1 ∗ (v 1 − p 1 )) 2 + (x 2 ∗ (v 2 − p 2 )) 2H(x) = { v ∈ R 2 |u(x, v) ≥ 0 }and exogenous restrictions for the initial feasible domain are given byX = { x ∈ R 2 |2x 1 = x 2}In the illustration E(v) = c ≠ p, such that graphically speaking: p is little above and to the leftfrom c. Let{}− r c (x) = min min min |v n − c n | ∣ u s (x, v) = 0n s vbe the radius of the cube enclosed by H(x) and+ r c (x) = maxnmax |v n − c n |v∈H(x)be the radius of the cube enclosing H(x). The left part of figure 1 illustrates the v-space andthe right part the x space. The lower left part of figure 1, illustrates the search for the largestenclosed cube. The inner cube is a subset of H(x I ) which intersects with the closure of H(x I )and thus corresponds to the global solution for the optimisation problem in the definitionof − r c (x I ). The other three larger cubes are partially outside H(x I ) and correspond to localsolutions for the optimisation problems in the definition of − r c (x I ). In figure 1, the happy setcorresponding point x III is enclosed by a cube with a radius identical to the radius of the cubebelonging to point x I : − r c (x I ) = + r c (x III ). Without knowing the details of the probabilityspace (V, V, Pr), the bounds guarantee that the probability mass of H(x III ) is smaller than orequal to the probability mass of H(x I ). Thus R(x I ) ≥ R(x III ). Therefore the feasible domain Xcan be reduced toX ′ := { x ∈ X|x 1 ≤ x1III }A second illustration of the R(x)-bounding methods, is the application in the context of conditionalMC sampling for estimating R(x). In the situation that the probability mass of shape− H(x) inside H(x) is known, then only samples outside − H(x) are needed to estimate R(x).The largest diamond is an example of such an enclosed shape: − D(c, − rd ∗ (x)) ⊆ H(x), with{ N∑}− rd ∗ (x) = min min |v n − c n |s v ∣ u s(x, v) = 0 . Consider that Robustness Programming techniquesexist which can compute − R(x) = Pr {v ∈ − D(c, − rd ∗ (x))} efficiently and cann=1generate